def render_bmf_space_webpage(field_label, level_label):
info = {}
t = "Bianchi modular forms of level %s over %s" % (level_label,
field_label)
credit = bianchi_credit
bread = [('Bianchi modular forms', url_for(".index")),
(field_pretty(field_label),
url_for(".render_bmf_field_dim_table_gl2",
field_label=field_label)), (level_label, '')]
friends = []
properties2 = []
if not field_label_regex.match(field_label):
info[
'err'] = "%s is not a valid label for an imaginary quadratic field" % field_label
else:
pretty_field_label = field_pretty(field_label)
if not db_dims().find({'field_label': field_label}):
info[
'err'] = "no dimension information exists in the database for field %s" % pretty_field_label
else:
t = "Bianchi Modular Forms of level %s over %s" % (
level_label, pretty_field_label)
data = db_dims().find({
'field_label': field_label,
'level_label': level_label
})
nres = data.count()
if nres == 0:
info[
'err'] = "no dimension information exists in the database for level %s and field %s" % (
level_label, pretty_field_label)
else:
data = data.next()
info['label'] = data['label']
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_label'] = field_label
info['pretty_field_label'] = pretty_field_label
info['level_label'] = level_label
info['level_norm'] = data['level_norm']
info['field_degree'] = nf.degree()
info['field_classno'] = nf.class_number()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info['field_knowl'] = nf_display_knowl(field_label,
getDBConnection(),
pretty_field_label)
w = 'i' if nf.disc() == -4 else 'a'
L = nf.K().change_names(w)
alpha = L.gen()
info['field_gen'] = latex(alpha)
I = ideal_from_label(L, level_label)
info['level_gen'] = latex(I.gens_reduced()[0])
info['level_fact'] = latex(I.factor())
dim_data = data['gl2_dims']
weights = dim_data.keys()
weights.sort(key=lambda w: int(w))
for w in weights:
dim_data[w]['dim'] = dim_data[w]['cuspidal_dim']
info['dim_data'] = dim_data
info['weights'] = weights
info['nweights'] = len(weights)
newdim = data['gl2_dims']['2']['new_dim']
newforms = db_forms().find({
'field_label': field_label,
'level_label': level_label
}).sort('label_suffix', ASCENDING)
info['nfdata'] = [{
'label':
f['short_label'],
'url':
url_for(".render_bmf_webpage",
field_label=f['field_label'],
level_label=f['level_label'],
label_suffix=f['label_suffix']),
'wt':
f['weight'],
'dim':
f['dimension'],
'sfe':
"+1" if f['sfe'] == 1 else "-1",
'bc':
bc_info(f['bc']),
'cm':
cm_info(f['CM']),
} for f in newforms]
info['nnewforms'] = len(info['nfdata'])
properties2 = [('Base field', pretty_field_label),
('Level', info['level_label']),
('Norm', str(info['level_norm'])),
('New dimension', str(newdim))]
friends = [('Newform {}'.format(f['label']), f['url'])
for f in info['nfdata']]
return render_template("bmf-space.html",
info=info,
credit=credit,
title=t,
bread=bread,
properties2=properties2,
friends=friends)
def FIELD(label):
nf = WebNumberField(label, gen_name=special_names.get(label, 'a'))
nf.parse_NFelt = lambda s: nf.K()([QQ(c.encode()) for c in s.split(",")])
nf.latex_poly = web_latex(nf.poly())
return nf
def bmf_field_dim_table(**args):
argsdict = to_dict(args)
argsdict.update(to_dict(request.args))
gl_or_sl = argsdict['gl_or_sl']
field_label = argsdict['field_label']
field_label = nf_string_to_label(field_label)
start = 0
if 'start' in argsdict:
start = int(argsdict['start'])
info = {}
info['gl_or_sl'] = gl_or_sl
# level_flag controls whether to list all levels ('all'), only
# those with positive cuspidal dimension ('cusp'), or only those
# with positive new dimension ('new'). Default is 'cusp'.
level_flag = argsdict.get('level_flag', 'cusp')
info['level_flag'] = level_flag
count = 50
if 'count' in argsdict:
count = int(argsdict['count'])
pretty_field_label = field_pretty(field_label)
bread = [('Bianchi Modular Forms', url_for(".index")),
(pretty_field_label, ' ')]
properties = []
if gl_or_sl == 'gl2_dims':
info['group'] = 'GL(2)'
info['bgroup'] = '\GL(2,\mathcal{O}_K)'
else:
info['group'] = 'SL(2)'
info['bgroup'] = '\SL(2,\mathcal{O}_K)'
t = ' '.join([
'Dimensions of spaces of {} Bianchi modular forms over'.format(
info['group']), pretty_field_label
])
query = {}
query['field_label'] = field_label
query[gl_or_sl] = {'$exists': True}
if level_flag != 'all':
# find which weights are present (TODO: get this from a stats collection)
wts = list(
sum((Set(d.keys()) for d in db_dims().distinct(gl_or_sl)), Set()))
if level_flag == 'cusp':
# restrict the query to only return levels where at least one
# cuspidal dimension is positive:
query.update({
'$or': [{
gl_or_sl + '.{}.cuspidal_dim'.format(w): {
'$gt': 0
}
} for w in wts]
})
if level_flag == 'new':
# restrict the query to only return levels where at least one
# new dimension is positive:
query.update({
'$or': [{
gl_or_sl + '.{}.new_dim'.format(w): {
'$gt': 0
}
} for w in wts]
})
data = db_dims().find(query)
data = data.sort([('level_norm', ASCENDING)])
info['number'] = nres = data.count()
if nres > count or start != 0:
info['report'] = 'Displaying items %s-%s of %s levels,' % (
start + 1, min(nres, start + count), nres)
else:
info['report'] = 'Displaying all %s levels,' % nres
data = list(data.skip(start).limit(count))
info['field'] = field_label
info['field_pretty'] = pretty_field_label
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
weights = set()
for dat in data:
weights = weights.union(set(dat[gl_or_sl].keys()))
weights = list([int(w) for w in weights])
weights.sort()
info['weights'] = weights
info['nweights'] = len(weights)
info['count'] = count
info['start'] = start
info['more'] = int(start + count < nres)
dims = {}
for dat in data:
dims[dat['level_label']] = d = {}
for w in weights:
sw = str(w)
if sw in dat[gl_or_sl]:
d[w] = {
'd': dat[gl_or_sl][sw]['cuspidal_dim'],
'n': dat[gl_or_sl][sw]['new_dim']
}
else:
d[w] = {'d': '?', 'n': '?'}
info['nlevels'] = len(data)
dimtable = [{
'level_label':
dat['level_label'],
'level_norm':
dat['level_norm'],
'level_space':
url_for(".render_bmf_space_webpage",
field_label=field_label,
level_label=dat['level_label'])
if gl_or_sl == 'gl2_dims' else "",
'dims':
dims[dat['level_label']]
} for dat in data]
print("Length of dimtable = {}".format(len(dimtable)))
info['dimtable'] = dimtable
return render_template("bmf-field_dim_table.html",
info=info,
title=t,
properties=properties,
bread=bread)
def render_field_webpage(args):
data = None
C = base.getDBConnection()
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), latex(data['conductor'].factor()))
data['galois_group'] = group_display_knowl(n, t, C)
data['cclasses'] = cclasses_display_knowl(n, t, C)
data['character_table'] = character_table_display_knowl(n, t, C)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
ram_primes = D.prime_factors()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
npr = len(ram_primes)
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['frob_data'], data['seeram'] = frobs(nf)
data['phrase'] = group_phrase(n, t, C)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'Not' in str(data['class_number']):
grh_lab=''
grh_label=''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
if nf.degree() > 1:
gpK = nf.gpK()
rootof1coeff = gpK.nfrootsof1()[2]
rootofunity = Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a'))
else:
rootofunity = Ra('-1')
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': web_latex(rootofunity),
'fund_units': nf.units(),
'grh_label': grh_label
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
info['learnmore'] = [('Global number field labels', url_for(
".render_labels_page")),
(Completename, url_for(".render_discriminants_page")),
('How data was computed', url_for(".how_computed_page"))]
if info['signature'] == [0,1]:
info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Label', label),
('Degree', '%s' % data['degree']),
('Signature', '$%s$' % data['signature']),
('Discriminant', '$%s$' % data['disc_factor']),
('Ramified ' + primes + '', '$%s$' % ram_primes),
('Class number', '%s %s' % (data['class_number'], grh_lab)),
('Class group', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group', group_display_short(data['degree'], t, C))
]
from lmfdb.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m==0: return ''
if m==1: return '*'
return '*<sup>%d</sup>'% m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
# del info['_id']
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), learnmore=info.pop('learnmore'), info=info)
def render_field_webpage(args):
data = None
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search Results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args[
'label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number_latex()
ram_primes = nf.ramified_primes()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars) > 0:
data['dirichlet_group'] = [
'<a href = "%s">$\chi_{%s}(%s,·)$</a>' %
(url_for('characters.render_Dirichletwebpage',
modulus=data['conductor'],
number=j), data['conductor'], j)
for j in dirichlet_chars
]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(
data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
factored_conductor = factor_base_factor(data['conductor'],
ram_primes)
factored_conductor = factor_base_factorization_latex(
factored_conductor)
data['conductor'] = "\(%s=%s\)" % (str(
data['conductor']), factored_conductor)
data['galois_group'] = group_display_knowl(n, t)
data['cclasses'] = cclasses_display_knowl(n, t)
data['character_table'] = character_table_display_knowl(n, t)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
data['frob_data'], data['seeram'] = frobs(nf)
# Bad prime information
npr = len(ram_primes)
ramified_algebras_data = nf.ramified_algebras_data()
if isinstance(ramified_algebras_data, str):
loc_alg = ''
else:
# [label, latex, e, f, c, gal]
loc_alg = ''
for j in range(npr):
if ramified_algebras_data[j] is None:
loc_alg += '<tr><td>%s<td colspan="7">Data not computed' % str(
ram_primes[j])
else:
mydat = ramified_algebras_data[j]
p = ram_primes[j]
loc_alg += '<tr><td rowspan="%d">$%s$</td>' % (len(mydat),
str(p))
mm = mydat[0]
myurl = url_for('local_fields.by_label', label=mm[0])
lab = mm[0]
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
for mm in mydat[1:]:
lab = mm[0]
if mm[3] * mm[2] == 1:
lab = r'$\Q_{%s}$' % str(p)
loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$' % (
myurl, lab, mm[1], mm[2], mm[3], mm[4], mm[5],
show_slope_content(mm[8], mm[6], mm[7]))
loc_alg += '</tbody></table>'
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['phrase'] = group_phrase(n, t)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh(
) else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'Not' in str(data['class_number']):
grh_lab = ''
grh_label = ''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
if nf.degree() > 1:
gpK = nf.gpK()
rootof1coeff = gpK.nfrootsof1()
rootofunityorder = int(rootof1coeff[1])
rootof1coeff = rootof1coeff[2]
rootofunity = web_latex(
Ra(
str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace(
'x', 'a')))
rootofunity += ' (order $%d$)' % rootofunityorder
else:
rootofunity = web_latex(Ra('-1')) + ' (order $2$)'
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': rootofunity,
'fund_units': nf.units(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet character group',
url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
resinfo = []
galois_closure = nf.galois_closure()
if galois_closure[0] > 0:
if len(galois_closure[1]) > 0:
resinfo.append(('gc', galois_closure[1]))
if len(galois_closure[2]) > 0:
info['friends'].append(('Galois closure',
url_for(".by_label",
label=galois_closure[2][0])))
else:
resinfo.append(('gc', [dnc]))
sextic_twins = nf.sextic_twin()
if sextic_twins[0] > 0:
if len(sextic_twins[1]) > 0:
resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
else:
resinfo.append(('sex', dnc))
siblings = nf.siblings()
# [degsib list, label list]
# first is list of [deg, num expected, list of knowls]
if len(siblings[0]) > 0:
for sibdeg in siblings[0]:
if len(sibdeg[2]) == 0:
sibdeg[2] = dnc
else:
sibdeg[2] = ', '.join(sibdeg[2])
if len(sibdeg[2]) < sibdeg[1]:
sibdeg[2] += ', some ' + dnc
resinfo.append(('sib', siblings[0]))
for lab in siblings[1]:
if lab != '':
labparts = lab.split('.')
info['friends'].append(("Degree %s sibling" % labparts[0],
url_for(".by_label", label=lab)))
arith_equiv = nf.arith_equiv()
if arith_equiv[0] > 0:
if len(arith_equiv[1]) > 0:
resinfo.append(
('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
for aelab in arith_equiv[2]:
info['friends'].append(('Arithmetically equivalent sibling',
url_for(".by_label", label=aelab)))
else:
resinfo.append(('ae', dnc, len(arith_equiv[1])))
info['resinfo'] = resinfo
learnmore = learnmore_list()
#if info['signature'] == [0,1]:
# info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Label', label), ('Degree', '$%s$' % data['degree']),
('Signature', '$%s$' % data['signature']),
('Discriminant', '$%s$' % data['disc_factor']),
('Ramified ' + primes + '', '$%s$' % ram_primes),
('Class number', '%s %s' % (data['class_number'], grh_lab)),
('Class group',
'%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group', group_display_short(data['degree'], t))]
downloads = []
for lang in [["Magma", "magma"], ["SageMath", "sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_code_download",
nf=label,
download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m == 0: return ''
if m == 1: return '*'
return '*<sup>%d</sup>' % m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
return render_template("number_field.html",
properties2=properties2,
credit=NF_credit,
title=title,
bread=bread,
code=nf.code,
friends=info.pop('friends'),
downloads=downloads,
learnmore=learnmore,
info=info)
def render_hmf_webpage(**args):
C = getDBConnection()
if 'data' in args:
data = args['data']
label = data['label']
else:
label = str(args['label'])
data = C.hmfs.forms.find_one({'label': label})
if data is None:
return "No such form"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
hmf_field = C.hmfs.fields.find_one({'label': data['field_label']})
gen_name = findvar(hmf_field['ideals'])
nf = WebNumberField(data['field_label'], gen_name=gen_name)
info['hmf_field'] = hmf_field
info['field'] = nf
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
info['downloads'] = [
('Download to Magma', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='magma')),
('Download to Sage', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='sage'))
]
if hmf_field['narrow_class_no'] == 1 and nf.disc()**2 * data['level_norm'] < 40000:
info['friends'] = [('L-function',
url_for("l_functions.l_function_hmf_page", field=info['field_label'], label=info['label'], character='0', number='0'))]
else:
info['friends'] = [('L-function not available', "")]
if data['dimension'] == 1: # Try to attach associated elliptic curve
lab = split_class_label(info['label'])
ec_from_hmf = db_ecnf().find_one({"label": label + '1'})
if ec_from_hmf == None:
info['friends'] += [('Elliptic curve not available', "")]
else:
info['friends'] += [('Isogeny class ' + info['label'], url_for("ecnf.show_ecnf_isoclass", nf=lab[0], conductor_label=lab[1], class_label=lab[2]))]
bread = [('Hilbert Modular Forms', url_for(".hilbert_modular_form_render_webpage")), ('%s' % data[
'label'], ' ')]
t = "Hilbert Cusp Form %s" % info['label']
forms_space = C.hmfs.forms.search.find(
{'field_label': data['field_label'], 'level_ideal': data['level_ideal']},{'dimension':True})
dim_space = 0
for v in forms_space:
dim_space += v['dimension']
info['newspace_dimension'] = dim_space
eigs = data['hecke_eigenvalues']
eigs = eigs[:min(len(eigs), numeigs)]
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{'eigenvalue': teXify_pol(eigs[i]),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = teXify_pol(info['hecke_polynomial'])
if 'AL_eigenvalues_fixed' in data:
if data['AL_eigenvalues_fixed'] == 'done':
info['AL_eigs'] = [{'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]} for al in data['AL_eigenvalues']]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
info['AL_eigs_count'] = len(info['AL_eigs']) != 0
max_eig_len = max([len(eig['eigenvalue']) for eig in info['eigs']])
display_eigs = display_eigs or (max_eig_len<=300)
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
for eig in info['AL_eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties2 = [('Base field', '%s' % info['field'].field_pretty()),
('Weight', '%s' % data['weight']),
('Level norm', '%s' % data['level_norm']),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Label', '%s' % data['label']),
('Dimension', '%s' % data['dimension']),
('CM', is_CM),
('Base change', is_base_change)
]
return render_template("hilbert_modular_form.html", downloads=info["downloads"], info=info, properties2=properties2, credit=hmf_credit, title=t, bread=bread, friends=info['friends'], learnmore=learnmore_list())
def bmf_field_dim_table(**args):
argsdict = to_dict(args)
argsdict.update(to_dict(request.args))
gl_or_sl = argsdict['gl_or_sl']
field_label=argsdict['field_label']
field_label = nf_string_to_label(field_label)
start = 0
if 'start' in argsdict:
start = int(argsdict['start'])
info={}
info['gl_or_sl'] = gl_or_sl
# level_flag controls whether to list all levels ('all'), only
# those with positive cuspidal dimension ('cusp'), or only those
# with positive new dimension ('new'). Default is 'cusp'.
level_flag = argsdict.get('level_flag', 'cusp')
info['level_flag'] = level_flag
count = 50
if 'count' in argsdict:
count = int(argsdict['count'])
pretty_field_label = field_pretty(field_label)
bread = [('Bianchi Modular Forms', url_for(".index")), (
pretty_field_label, ' ')]
properties = []
if gl_or_sl=='gl2_dims':
info['group'] = 'GL(2)'
info['bgroup'] = '\GL(2,\mathcal{O}_K)'
else:
info['group'] = 'SL(2)'
info['bgroup'] = '\SL(2,\mathcal{O}_K)'
t = ' '.join(['Dimensions of Spaces of {} Bianchi Modular Forms over'.format(info['group']), pretty_field_label])
query = {}
query['field_label'] = field_label
query[gl_or_sl] = {'$exists': True}
data = db_dims().find(query)
data = data.sort([('level_norm', ASCENDING)])
info['number'] = nres = data.count()
if nres > count or start != 0:
info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'Displaying all %s levels,' % nres
# convert data to a list and eliminate levels where all
# new/cuspidal dimensions are 0. (This could be done at the
# search stage, but that requires adding new fields to each
# record.)
def filter(dat, flag):
dat1 = dat[gl_or_sl]
return any([int(dat1[w][flag])>0 for w in dat1])
flag = 'cuspidal_dim' if level_flag=='cusp' else 'new_dim'
data = [dat for dat in data if level_flag == 'all' or filter(dat, flag)]
data = data[start:start+count]
info['field'] = field_label
info['field_pretty'] = pretty_field_label
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
weights = set()
for dat in data:
weights = weights.union(set(dat[gl_or_sl].keys()))
weights = list([int(w) for w in weights])
weights.sort()
info['weights'] = weights
info['nweights'] = len(weights)
info['count'] = count
info['start'] = start
info['more'] = int(start + count < nres)
data.sort(key = lambda x: [int(y) for y in x['level_label'].split(".")])
dims = {}
for dat in data:
dims[dat['level_label']] = d = {}
for w in weights:
sw = str(w)
if sw in dat[gl_or_sl]:
d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
'n': dat[gl_or_sl][sw]['new_dim']}
else:
d[w] = {'d': '?', 'n': '?'}
info['nlevels'] = len(data)
dimtable = [{'level_label': dat['level_label'],
'level_norm': dat['level_norm'],
'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
'dims': dims[dat['level_label']]} for dat in data]
info['dimtable'] = dimtable
return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
def FIELD(label):
nf = WebNumberField(label, gen_name=special_names.get(label, 'a'))
nf.parse_NFelt = lambda s: nf.K()([QQ(c.encode()) for c in s.split(",")])
nf.latex_poly = web_latex(nf.poly())
return nf
def render_field_webpage(args):
data = None
info = {}
bread = [('Global Number Fields', url_for(".number_field_render_webpage"))]
# This function should not be called unless label is set.
label = clean_input(args['label'])
nf = WebNumberField(label)
data = {}
if nf.is_null():
bread.append(('Search Results', ' '))
info['err'] = 'There is no field with label %s in the database' % label
info['label'] = args['label_orig'] if 'label_orig' in args else args['label']
return search_input_error(info, bread)
info['wnf'] = nf
data['degree'] = nf.degree()
data['class_number'] = nf.class_number_latex()
ram_primes = nf.ramified_primes()
t = nf.galois_t()
n = nf.degree()
data['is_galois'] = nf.is_galois()
data['is_abelian'] = nf.is_abelian()
if nf.is_abelian():
conductor = nf.conductor()
data['conductor'] = conductor
dirichlet_chars = nf.dirichlet_group()
if len(dirichlet_chars)>0:
data['dirichlet_group'] = ['<a href = "%s">$\chi_{%s}(%s,·)$</a>' % (url_for('characters.render_Dirichletwebpage',modulus=data['conductor'], number=j), data['conductor'], j) for j in dirichlet_chars]
data['dirichlet_group'] = r'$\lbrace$' + ', '.join(data['dirichlet_group']) + r'$\rbrace$'
if data['conductor'].is_prime() or data['conductor'] == 1:
data['conductor'] = "\(%s\)" % str(data['conductor'])
else:
factored_conductor = factor_base_factor(data['conductor'], ram_primes)
factored_conductor = factor_base_factorization_latex(factored_conductor)
data['conductor'] = "\(%s=%s\)" % (str(data['conductor']), factored_conductor)
data['galois_group'] = group_display_knowl(n, t)
data['cclasses'] = cclasses_display_knowl(n, t)
data['character_table'] = character_table_display_knowl(n, t)
data['class_group'] = nf.class_group()
data['class_group_invs'] = nf.class_group_invariants()
data['signature'] = nf.signature()
data['coefficients'] = nf.coeffs()
nf.make_code_snippets()
D = nf.disc()
data['disc_factor'] = nf.disc_factored_latex()
if D.abs().is_prime() or D == 1:
data['discriminant'] = "\(%s\)" % str(D)
else:
data['discriminant'] = "\(%s=%s\)" % (str(D), data['disc_factor'])
data['frob_data'], data['seeram'] = frobs(nf)
# Bad prime information
npr = len(ram_primes)
ramified_algebras_data = nf.ramified_algebras_data()
if isinstance(ramified_algebras_data,str):
loc_alg = ''
else:
# [label, latex, e, f, c, gal]
loc_alg = ''
for j in range(npr):
if ramified_algebras_data[j] is None:
loc_alg += '<tr><td>%s<td colspan="7">Data not computed'%str(ram_primes[j])
else:
mydat = ramified_algebras_data[j]
p = ram_primes[j]
loc_alg += '<tr><td rowspan="%d">$%s$</td>'%(len(mydat),str(p))
mm = mydat[0]
myurl = url_for('local_fields.by_label', label=mm[0])
lab = mm[0]
if mm[3]*mm[2]==1:
lab = r'$\Q_{%s}$'%str(p)
loc_alg += '<td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$'%(myurl,lab,mm[1],mm[2],mm[3],mm[4],mm[5],show_slope_content(mm[8],mm[6],mm[7]))
for mm in mydat[1:]:
lab = mm[0]
if mm[3]*mm[2]==1:
lab = r'$\Q_{%s}$'%str(p)
loc_alg += '<tr><td><a href="%s">%s</a><td>$%s$<td>$%d$<td>$%d$<td>$%d$<td>%s<td>$%s$'%(myurl,lab,mm[1],mm[2],mm[3],mm[4],mm[5],show_slope_content(mm[8],mm[6],mm[7]))
loc_alg += '</tbody></table>'
ram_primes = str(ram_primes)[1:-1]
if ram_primes == '':
ram_primes = r'\textrm{None}'
data['phrase'] = group_phrase(n, t)
zk = nf.zk()
Ra = PolynomialRing(QQ, 'a')
zk = [latex(Ra(x)) for x in zk]
zk = ['$%s$' % x for x in zk]
zk = ', '.join(zk)
grh_label = '<small>(<a title="assuming GRH" knowl="nf.assuming_grh">assuming GRH</a>)</small>' if nf.used_grh() else ''
# Short version for properties
grh_lab = nf.short_grh_string()
if 'Not' in str(data['class_number']):
grh_lab=''
grh_label=''
pretty_label = field_pretty(label)
if label != pretty_label:
pretty_label = "%s: %s" % (label, pretty_label)
info.update(data)
if nf.degree() > 1:
gpK = nf.gpK()
rootof1coeff = gpK.nfrootsof1()
rootofunityorder = int(rootof1coeff[1])
rootof1coeff = rootof1coeff[2]
rootofunity = web_latex(Ra(str(pari("lift(%s)" % gpK.nfbasistoalg(rootof1coeff))).replace('x','a')))
rootofunity += ' (order $%d$)' % rootofunityorder
else:
rootofunity = web_latex(Ra('-1'))+ ' (order $2$)'
info.update({
'label': pretty_label,
'label_raw': label,
'polynomial': web_latex_split_on_pm(nf.poly()),
'ram_primes': ram_primes,
'integral_basis': zk,
'regulator': web_latex(nf.regulator()),
'unit_rank': nf.unit_rank(),
'root_of_unity': rootofunity,
'fund_units': nf.units(),
'grh_label': grh_label,
'loc_alg': loc_alg
})
bread.append(('%s' % info['label_raw'], ' '))
info['downloads_visible'] = True
info['downloads'] = [('worksheet', '/')]
info['friends'] = []
if nf.can_class_number():
# hide ones that take a lond time to compute on the fly
# note that the first degree 4 number field missed the zero of the zeta function
if abs(D**n) < 50000000:
info['friends'].append(('L-function', "/L/NumberField/%s" % label))
info['friends'].append(('Galois group', "/GaloisGroup/%dT%d" % (n, t)))
if 'dirichlet_group' in info:
info['friends'].append(('Dirichlet character group', url_for("characters.dirichlet_group_table",
modulus=int(conductor),
char_number_list=','.join(
[str(a) for a in dirichlet_chars]),
poly=info['polynomial'])))
resinfo=[]
galois_closure = nf.galois_closure()
if galois_closure[0]>0:
if len(galois_closure[1])>0:
resinfo.append(('gc', galois_closure[1]))
if len(galois_closure[2]) > 0:
info['friends'].append(('Galois closure',url_for(".by_label", label=galois_closure[2][0])))
else:
resinfo.append(('gc', [dnc]))
sextic_twins = nf.sextic_twin()
if sextic_twins[0]>0:
if len(sextic_twins[1])>0:
resinfo.append(('sex', r' $\times$ '.join(sextic_twins[1])))
else:
resinfo.append(('sex', dnc))
siblings = nf.siblings()
# [degsib list, label list]
# first is list of [deg, num expected, list of knowls]
if len(siblings[0])>0:
for sibdeg in siblings[0]:
if len(sibdeg[2]) ==0:
sibdeg[2] = dnc
else:
sibdeg[2] = ', '.join(sibdeg[2])
if len(sibdeg[2])<sibdeg[1]:
sibdeg[2] += ', some '+dnc
resinfo.append(('sib', siblings[0]))
for lab in siblings[1]:
if lab != '':
labparts = lab.split('.')
info['friends'].append(("Degree %s sibling"%labparts[0] ,url_for(".by_label", label=lab)))
arith_equiv = nf.arith_equiv()
if arith_equiv[0]>0:
if len(arith_equiv[1])>0:
resinfo.append(('ae', ', '.join(arith_equiv[1]), len(arith_equiv[1])))
for aelab in arith_equiv[2]:
info['friends'].append(('Arithmetically equivalent sibling',url_for(".by_label", label=aelab)))
else:
resinfo.append(('ae', dnc, len(arith_equiv[1])))
info['resinfo'] = resinfo
learnmore = learnmore_list()
#if info['signature'] == [0,1]:
# info['learnmore'].append(('Quadratic imaginary class groups', url_for(".render_class_group_data")))
# With Galois group labels, probably not needed here
# info['learnmore'] = [('Global number field labels',
# url_for(".render_labels_page")), ('Galois group
# labels',url_for(".render_groups_page")),
# (Completename,url_for(".render_discriminants_page"))]
title = "Global Number Field %s" % info['label']
if npr == 1:
primes = 'prime'
else:
primes = 'primes'
properties2 = [('Label', label),
('Degree', '$%s$' % data['degree']),
('Signature', '$%s$' % data['signature']),
('Discriminant', '$%s$' % data['disc_factor']),
('Ramified ' + primes + '', '$%s$' % ram_primes),
('Class number', '%s %s' % (data['class_number'], grh_lab)),
('Class group', '%s %s' % (data['class_group_invs'], grh_lab)),
('Galois Group', group_display_short(data['degree'], t))
]
downloads = []
for lang in [["Magma","magma"], ["SageMath","sage"], ["Pari/GP", "gp"]]:
downloads.append(('Download {} code'.format(lang[0]),
url_for(".nf_code_download", nf=label, download_type=lang[1])))
from lmfdb.artin_representations.math_classes import NumberFieldGaloisGroup
try:
info["tim_number_field"] = NumberFieldGaloisGroup(nf._data['coeffs'])
v = nf.factor_perm_repn(info["tim_number_field"])
def dopow(m):
if m==0: return ''
if m==1: return '*'
return '*<sup>%d</sup>'% m
info["mydecomp"] = [dopow(x) for x in v]
except AttributeError:
pass
return render_template("number_field.html", properties2=properties2, credit=NF_credit, title=title, bread=bread, code=nf.code, friends=info.pop('friends'), downloads=downloads, learnmore=learnmore, info=info)
def bmf_field_dim_table(**args):
argsdict = to_dict(args)
argsdict.update(to_dict(request.args))
gl_or_sl = argsdict['gl_or_sl']
field_label=argsdict['field_label']
field_label = nf_string_to_label(field_label)
start = 0
if 'start' in argsdict:
start = int(argsdict['start'])
info={}
info['gl_or_sl'] = gl_or_sl
# level_flag controls whether to list all levels ('all'), only
# those with positive cuspidal dimension ('cusp'), or only those
# with positive new dimension ('new'). Default is 'cusp'.
level_flag = argsdict.get('level_flag', 'cusp')
info['level_flag'] = level_flag
count = 50
if 'count' in argsdict:
count = int(argsdict['count'])
pretty_field_label = field_pretty(field_label)
bread = [('Bianchi Modular Forms', url_for(".index")), (
pretty_field_label, ' ')]
properties = []
if gl_or_sl=='gl2_dims':
info['group'] = 'GL(2)'
info['bgroup'] = '\GL(2,\mathcal{O}_K)'
else:
info['group'] = 'SL(2)'
info['bgroup'] = '\SL(2,\mathcal{O}_K)'
t = ' '.join(['Dimensions of spaces of {} Bianchi modular forms over'.format(info['group']), pretty_field_label])
query = {}
query['field_label'] = field_label
query[gl_or_sl] = {'$exists': True}
data = db_dims().find(query)
data = data.sort([('level_norm', ASCENDING)])
info['number'] = nres = data.count()
if nres > count or start != 0:
info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'Displaying all %s levels,' % nres
# convert data to a list and eliminate levels where all
# new/cuspidal dimensions are 0. (This could be done at the
# search stage, but that requires adding new fields to each
# record.)
def filter(dat, flag):
dat1 = dat[gl_or_sl]
return any([int(dat1[w][flag])>0 for w in dat1])
flag = 'cuspidal_dim' if level_flag=='cusp' else 'new_dim'
data = [dat for dat in data if level_flag == 'all' or filter(dat, flag)]
data = data[start:start+count]
info['field'] = field_label
info['field_pretty'] = pretty_field_label
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
weights = set()
for dat in data:
weights = weights.union(set(dat[gl_or_sl].keys()))
weights = list([int(w) for w in weights])
weights.sort()
info['weights'] = weights
info['nweights'] = len(weights)
info['count'] = count
info['start'] = start
info['more'] = int(start + count < nres)
data.sort(key = lambda x: [int(y) for y in x['level_label'].split(".")])
dims = {}
for dat in data:
dims[dat['level_label']] = d = {}
for w in weights:
sw = str(w)
if sw in dat[gl_or_sl]:
d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
'n': dat[gl_or_sl][sw]['new_dim']}
else:
d[w] = {'d': '?', 'n': '?'}
info['nlevels'] = len(data)
dimtable = [{'level_label': dat['level_label'],
'level_norm': dat['level_norm'],
'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
'dims': dims[dat['level_label']]} for dat in data]
info['dimtable'] = dimtable
return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
def render_hmf_webpage(**args):
C = getDBConnection()
if 'data' in args:
data = args['data']
label = data['label']
else:
label = str(args['label'])
data = C.hmfs.forms.find_one({'label': label})
if data is None:
return "No such form"
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
hmf_field = C.hmfs.fields.find_one({'label': data['field_label']})
gen_name = findvar(hmf_field['ideals'])
nf = WebNumberField(data['field_label'], gen_name=gen_name)
info['hmf_field'] = hmf_field
info['field'] = nf
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
info['downloads'] = [('Download to Magma',
url_for(".render_hmf_webpage_download",
field_label=info['field_label'],
label=info['label'],
download_type='magma')),
('Download to Sage',
url_for(".render_hmf_webpage_download",
field_label=info['field_label'],
label=info['label'],
download_type='sage'))]
if hmf_field['narrow_class_no'] == 1 and nf.disc(
)**2 * data['level_norm'] < 40000:
info['friends'] = [('L-function',
url_for("l_functions.l_function_hmf_page",
field=info['field_label'],
label=info['label'],
character='0',
number='0'))]
else:
info['friends'] = [('L-function not available', "")]
if data['dimension'] == 1: # Try to attach associated elliptic curve
lab = split_class_label(info['label'])
ec_from_hmf = db_ecnf().find_one({"label": label + '1'})
if ec_from_hmf == None:
info['friends'] += [('Elliptic curve not available', "")]
else:
info['friends'] += [('Isogeny class ' + info['label'],
url_for("ecnf.show_ecnf_isoclass",
nf=lab[0],
conductor_label=lab[1],
class_label=lab[2]))]
bread = [('Hilbert Modular Forms',
url_for(".hilbert_modular_form_render_webpage")),
('%s' % data['label'], ' ')]
t = "Hilbert Cusp Form %s" % info['label']
forms_space = C.hmfs.forms.search.find(
{
'field_label': data['field_label'],
'level_ideal': data['level_ideal']
}, {'dimension': True})
dim_space = 0
for v in forms_space:
dim_space += v['dimension']
info['newspace_dimension'] = dim_space
eigs = data['hecke_eigenvalues']
eigs = eigs[:min(len(eigs), numeigs)]
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{
'eigenvalue': teXify_pol(eigs[i]),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]
} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = teXify_pol(info['hecke_polynomial'])
if 'AL_eigenvalues_fixed' in data:
if data['AL_eigenvalues_fixed'] == 'done':
info['AL_eigs'] = [{
'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]
} for al in data['AL_eigenvalues']]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
else:
info['AL_eigs'] = [{'eigenvalue': '?', 'prime_ideal': '?'}]
info['AL_eigs_count'] = len(info['AL_eigs']) != 0
max_eig_len = max([len(eig['eigenvalue']) for eig in info['eigs']])
display_eigs = display_eigs or (max_eig_len <= 300)
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
for eig in info['AL_eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties2 = [('Base field', '%s' % info['field'].field_pretty()),
('Weight', '%s' % data['weight']),
('Level norm', '%s' % data['level_norm']),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Label', '%s' % data['label']),
('Dimension', '%s' % data['dimension']), ('CM', is_CM),
('Base change', is_base_change)]
return render_template("hilbert_modular_form.html",
downloads=info["downloads"],
info=info,
properties2=properties2,
credit=hmf_credit,
title=t,
bread=bread,
friends=info['friends'],
learnmore=learnmore_list())
def render_hmf_webpage(**args):
if 'data' in args:
data = args['data']
label = data['label']
else:
label = str(args['label'])
data = get_hmf(label)
if data is None:
flash(
Markup(
"Error: <span style='color:black'>%s</span> is not a valid Hilbert modular form label. It must be of the form (number field label) - (level label) - (orbit label) separated by dashes, such as 2.2.5.1-31.1-a"
% args['label']), "error")
return search_input_error()
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
hmf_field = get_hmf_field(data['field_label'])
gen_name = findvar(hmf_field['ideals'])
nf = WebNumberField(data['field_label'], gen_name=gen_name)
info['hmf_field'] = hmf_field
info['field'] = nf
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
info['downloads'] = [('Download to Magma',
url_for(".render_hmf_webpage_download",
field_label=info['field_label'],
label=info['label'],
download_type='magma')),
('Download to Sage',
url_for(".render_hmf_webpage_download",
field_label=info['field_label'],
label=info['label'],
download_type='sage'))]
if hmf_field['narrow_class_no'] == 1 and nf.disc(
)**2 * data['level_norm'] < 40000:
info['friends'] = [('L-function',
url_for("l_functions.l_function_hmf_page",
field=info['field_label'],
label=info['label'],
character='0',
number='0'))]
else:
info['friends'] = [('L-function not available', "")]
if data['dimension'] == 1: # Try to attach associated elliptic curve
lab = split_class_label(info['label'])
ec_from_hmf = db_ecnf().find_one({"label": label + '1'})
if ec_from_hmf == None:
info['friends'] += [('Elliptic curve not available', "")]
else:
info['friends'] += [('Isogeny class ' + info['label'],
url_for("ecnf.show_ecnf_isoclass",
nf=lab[0],
conductor_label=lab[1],
class_label=lab[2]))]
bread = [('Hilbert Modular Forms',
url_for(".hilbert_modular_form_render_webpage")),
('%s' % data['label'], ' ')]
t = "Hilbert Cusp Form %s" % info['label']
forms_space = db_search().find(
{
'field_label': data['field_label'],
'level_ideal': data['level_ideal']
}, {'dimension': True})
dim_space = 0
for v in forms_space:
dim_space += v['dimension']
info['newspace_dimension'] = dim_space
# Get hecke_polynomial, hecke_eigenvalues and AL_eigenvalues
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
info['numeigs'] = numeigs
hecke_pol = data['hecke_polynomial']
eigs = data['hecke_eigenvalues']
eigs = eigs[:min(len(eigs), numeigs)]
AL_eigs = data['AL_eigenvalues']
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{
'eigenvalue': teXify_pol(eigs[i]),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]
} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = web_latex_split_on_pm(teXify_pol(hecke_pol))
if not AL_eigs: # empty list
if data['level_norm'] == 1: # OK, no bad primes
info['AL_eigs'] = 'none'
else: # not OK, AL eigs are missing
info['AL_eigs'] = 'missing'
else:
info['AL_eigs'] = [{
'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]
} for al in data['AL_eigenvalues']]
max_eig_len = max([len(eig['eigenvalue']) for eig in info['eigs']])
display_eigs = display_eigs or (max_eig_len <= 300)
info['display_eigs'] = display_eigs
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties2 = [('Base field', '%s' % info['field'].field_pretty()),
('Weight', '%s' % data['weight']),
('Level norm', '%s' % data['level_norm']),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Label', '%s' % data['label']),
('Dimension', '%s' % data['dimension']), ('CM', is_CM),
('Base change', is_base_change)]
return render_template("hilbert_modular_form.html",
downloads=info["downloads"],
info=info,
properties2=properties2,
credit=hmf_credit,
title=t,
bread=bread,
friends=info['friends'],
learnmore=learnmore_list())
def FIELD(label):
nf = WebNumberField(label, gen_name=special_names.get(label, "a"))
nf.parse_NFelt = lambda s: nf.K()([QQ(str(c)) for c in s])
nf.latex_poly = web_latex(nf.poly())
return nf
def render_hmf_webpage(**args):
if 'data' in args:
data = args['data']
label = data['label']
else:
label = str(args['label'])
data = get_hmf(label)
if data is None:
flash(Markup("Error: <span style='color:black'>%s</span> is not a valid Hilbert modular form label. It must be of the form (number field label) - (level label) - (orbit label) separated by dashes, such as 2.2.5.1-31.1-a" % args['label']), "error")
return search_input_error()
info = {}
try:
info['count'] = args['count']
except KeyError:
info['count'] = 10
hmf_field = get_hmf_field(data['field_label'])
gen_name = findvar(hmf_field['ideals'])
nf = WebNumberField(data['field_label'], gen_name=gen_name)
info['hmf_field'] = hmf_field
info['field'] = nf
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info.update(data)
info['downloads'] = [
('Download to Magma', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='magma')),
('Download to Sage', url_for(".render_hmf_webpage_download", field_label=info['field_label'], label=info['label'], download_type='sage'))
]
if hmf_field['narrow_class_no'] == 1 and nf.disc()**2 * data['level_norm'] < 40000:
info['friends'] = [('L-function',
url_for("l_functions.l_function_hmf_page", field=info['field_label'], label=info['label'], character='0', number='0'))]
else:
info['friends'] = [('L-function not available', "")]
if data['dimension'] == 1: # Try to attach associated elliptic curve
lab = split_class_label(info['label'])
ec_from_hmf = db_ecnf().find_one({"label": label + '1'})
if ec_from_hmf == None:
info['friends'] += [('Elliptic curve not available', "")]
else:
info['friends'] += [('Isogeny class ' + info['label'], url_for("ecnf.show_ecnf_isoclass", nf=lab[0], conductor_label=lab[1], class_label=lab[2]))]
bread = [("Modular Forms", url_for('mf.modular_form_main_page')), ('Hilbert Modular Forms', url_for(".hilbert_modular_form_render_webpage")),
('%s' % data['label'], ' ')]
t = "Hilbert Cusp Form %s" % info['label']
forms_space = db_forms().find(
{'field_label': data['field_label'], 'level_ideal': data['level_ideal']},{'dimension':True})
dim_space = 0
for v in forms_space:
dim_space += v['dimension']
info['newspace_dimension'] = dim_space
# Get hecke_polynomial, hecke_eigenvalues and AL_eigenvalues
try:
numeigs = request.args['numeigs']
numeigs = int(numeigs)
except:
numeigs = 20
info['numeigs'] = numeigs
hecke_pol = data['hecke_polynomial']
eigs = data['hecke_eigenvalues']
eigs = eigs[:min(len(eigs), numeigs)]
AL_eigs = data['AL_eigenvalues']
primes = hmf_field['primes']
n = min(len(eigs), len(primes))
info['eigs'] = [{'eigenvalue': add_space_if_positive(teXify_pol(eigs[i])),
'prime_ideal': teXify_pol(primes[i]),
'prime_norm': primes[i][1:primes[i].index(',')]} for i in range(n)]
try:
display_eigs = request.args['display_eigs']
if display_eigs in ['True', 'true', '1', 'yes']:
display_eigs = True
else:
display_eigs = False
except KeyError:
display_eigs = False
if 'numeigs' in request.args:
display_eigs = True
info['hecke_polynomial'] = web_latex_split_on_pm(teXify_pol(hecke_pol))
if not AL_eigs: # empty list
if data['level_norm']==1: # OK, no bad primes
info['AL_eigs'] = 'none'
else: # not OK, AL eigs are missing
info['AL_eigs'] = 'missing'
else:
info['AL_eigs'] = [{'eigenvalue': teXify_pol(al[1]),
'prime_ideal': teXify_pol(al[0]),
'prime_norm': al[0][1:al[0].index(',')]} for al in data['AL_eigenvalues']]
max_eig_len = max([len(eig['eigenvalue']) for eig in info['eigs']])
display_eigs = display_eigs or (max_eig_len<=300)
info['display_eigs'] = display_eigs
if not display_eigs:
for eig in info['eigs']:
if len(eig['eigenvalue']) > 300:
eig['eigenvalue'] = '...'
info['level_ideal'] = teXify_pol(info['level_ideal'])
if 'is_CM' in data:
is_CM = data['is_CM']
else:
is_CM = '?'
info['is_CM'] = is_CM
if 'is_base_change' in data:
is_base_change = data['is_base_change']
else:
is_base_change = '?'
info['is_base_change'] = is_base_change
if 'q_expansions' in data:
info['q_expansions'] = data['q_expansions']
properties2 = [('Base field', '%s' % info['field'].field_pretty()),
('Weight', '%s' % data['weight']),
('Level norm', '%s' % data['level_norm']),
('Level', '$' + teXify_pol(data['level_ideal']) + '$'),
('Label', '%s' % data['label']),
('Dimension', '%s' % data['dimension']),
('CM', is_CM),
('Base change', is_base_change)
]
return render_template("hilbert_modular_form.html", downloads=info["downloads"], info=info, properties2=properties2, credit=hmf_credit, title=t, bread=bread, friends=info['friends'], learnmore=learnmore_list())
def render_bmf_space_webpage(field_label, level_label):
info = {}
t = "Bianchi modular forms of level %s over %s" % (level_label, field_label)
credit = bianchi_credit
bread = [('Bianchi modular forms', url_for(".index")),
(field_pretty(field_label), url_for(".render_bmf_field_dim_table_gl2", field_label=field_label)),
(level_label, '')]
friends = []
properties2 = []
if not field_label_regex.match(field_label):
info['err'] = "%s is not a valid label for an imaginary quadratic field" % field_label
else:
pretty_field_label = field_pretty(field_label)
if not db_dims().find({'field_label': field_label}):
info['err'] = "no dimension information exists in the database for field %s" % pretty_field_label
else:
t = "Bianchi Modular Forms of level %s over %s" % (level_label, pretty_field_label)
data = db_dims().find({'field_label': field_label, 'level_label': level_label})
nres = data.count()
if nres==0:
info['err'] = "no dimension information exists in the database for level %s and field %s" % (level_label, pretty_field_label)
else:
data = data.next()
info['label'] = data['label']
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_label'] = field_label
info['pretty_field_label'] = pretty_field_label
info['level_label'] = level_label
info['level_norm'] = data['level_norm']
info['field_degree'] = nf.degree()
info['field_classno'] = nf.class_number()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
info['field_knowl'] = nf_display_knowl(field_label, getDBConnection(), pretty_field_label)
w = 'i' if nf.disc()==-4 else 'a'
L = nf.K().change_names(w)
alpha = L.gen()
info['field_gen'] = latex(alpha)
I = ideal_from_label(L,level_label)
info['level_gen'] = latex(I.gens_reduced()[0])
info['level_fact'] = latex(I.factor())
dim_data = data['gl2_dims']
weights = dim_data.keys()
weights.sort(key=lambda w: int(w))
for w in weights:
dim_data[w]['dim']=dim_data[w]['cuspidal_dim']
info['dim_data'] = dim_data
info['weights'] = weights
info['nweights'] = len(weights)
newdim = data['gl2_dims']['2']['new_dim']
newforms = db_forms().find({'field_label':field_label, 'level_label':level_label}).sort('label_suffix', ASCENDING)
info['nfdata'] = [{
'label': f['short_label'],
'url': url_for(".render_bmf_webpage",field_label=f['field_label'], level_label=f['level_label'], label_suffix=f['label_suffix']),
'wt': f['weight'],
'dim': f['dimension'],
'sfe': "+1" if f['sfe']==1 else "-1",
'bc': bc_info(f['bc']),
'cm': cm_info(f['CM']),
} for f in newforms]
info['nnewforms'] = len(info['nfdata'])
properties2 = [('Base field', pretty_field_label), ('Level',info['level_label']), ('Norm',str(info['level_norm'])), ('New dimension',str(newdim))]
friends = [('Newform {}'.format(f['label']), f['url']) for f in info['nfdata'] ]
return render_template("bmf-space.html", info=info, credit=credit, title=t, bread=bread, properties2=properties2, friends=friends)
def bmf_field_dim_table(**args):
argsdict = to_dict(args)
argsdict.update(to_dict(request.args))
gl_or_sl = argsdict['gl_or_sl']
field_label=argsdict['field_label']
field_label = nf_string_to_label(field_label)
start = 0
if 'start' in argsdict:
start = int(argsdict['start'])
info={}
info['gl_or_sl'] = gl_or_sl
# level_flag controls whether to list all levels ('all'), only
# those with positive cuspidal dimension ('cusp'), or only those
# with positive new dimension ('new'). Default is 'cusp'.
level_flag = argsdict.get('level_flag', 'cusp')
info['level_flag'] = level_flag
count = 50
if 'count' in argsdict:
count = int(argsdict['count'])
pretty_field_label = field_pretty(field_label)
bread = [('Bianchi Modular Forms', url_for(".index")), (
pretty_field_label, ' ')]
properties = []
if gl_or_sl=='gl2_dims':
info['group'] = 'GL(2)'
info['bgroup'] = '\GL(2,\mathcal{O}_K)'
else:
info['group'] = 'SL(2)'
info['bgroup'] = '\SL(2,\mathcal{O}_K)'
t = ' '.join(['Dimensions of spaces of {} Bianchi modular forms over'.format(info['group']), pretty_field_label])
query = {}
query['field_label'] = field_label
query[gl_or_sl] = {'$exists': True}
if level_flag != 'all':
# find which weights are present (TODO: get this from a stats collection)
wts = list(sum((Set(d.keys()) for d in db_dims().distinct(gl_or_sl)),Set()))
if level_flag == 'cusp':
# restrict the query to only return levels where at least one
# cuspidal dimension is positive:
query.update(
{'$or':[{gl_or_sl+'.{}.cuspidal_dim'.format(w):{'$gt':0}} for w in wts]}
)
if level_flag == 'new':
# restrict the query to only return levels where at least one
# new dimension is positive:
query.update(
{'$or':[{gl_or_sl+'.{}.new_dim'.format(w):{'$gt':0}} for w in wts]}
)
data = db_dims().find(query)
data = data.sort([('level_norm', ASCENDING)])
info['number'] = nres = data.count()
if nres > count or start != 0:
info['report'] = 'Displaying items %s-%s of %s levels,' % (start + 1, min(nres, start + count), nres)
else:
info['report'] = 'Displaying all %s levels,' % nres
data = list(data.skip(start).limit(count))
info['field'] = field_label
info['field_pretty'] = pretty_field_label
nf = WebNumberField(field_label)
info['base_galois_group'] = nf.galois_string()
info['field_degree'] = nf.degree()
info['field_disc'] = str(nf.disc())
info['field_poly'] = teXify_pol(str(nf.poly()))
weights = set()
for dat in data:
weights = weights.union(set(dat[gl_or_sl].keys()))
weights = list([int(w) for w in weights])
weights.sort()
info['weights'] = weights
info['nweights'] = len(weights)
info['count'] = count
info['start'] = start
info['more'] = int(start + count < nres)
dims = {}
for dat in data:
dims[dat['level_label']] = d = {}
for w in weights:
sw = str(w)
if sw in dat[gl_or_sl]:
d[w] = {'d': dat[gl_or_sl][sw]['cuspidal_dim'],
'n': dat[gl_or_sl][sw]['new_dim']}
else:
d[w] = {'d': '?', 'n': '?'}
info['nlevels'] = len(data)
dimtable = [{'level_label': dat['level_label'],
'level_norm': dat['level_norm'],
'level_space': url_for(".render_bmf_space_webpage", field_label=field_label, level_label=dat['level_label']) if gl_or_sl=='gl2_dims' else "",
'dims': dims[dat['level_label']]} for dat in data]
print("Length of dimtable = {}".format(len(dimtable)))
info['dimtable'] = dimtable
return render_template("bmf-field_dim_table.html", info=info, title=t, properties=properties, bread=bread)
